The values of $x$ for which the angle between the vectors $x^2 \hat{i} + 2 x \hat{j} + \hat{k}$ and $\hat{i} - 2 \hat{j} + x \hat{k}$ is obtuse,lie in the interval

The component of $\vec{i} + \vec{j}$ along $\vec{j} + \vec{k}$ is:

If $|\vec{a}|=1, |\vec{b}|=2, |\vec{a}-\vec{b}|^2+|\vec{a}+2\vec{b}|^2=20$, then the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is:

If $\overrightarrow{a}=\hat{i}-\hat{j}-\hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}-3 \hat{j}+\hat{k}$ and the orthogonal projection of $\overrightarrow{b}$ on $\overrightarrow{a}$ is $\frac{4}{3}(\hat{i}-\hat{j}-\hat{k})$,then $\lambda$ is equal to

Suppose $\overrightarrow{a}=\lambda \hat{i}-7 \hat{j}+3 \hat{k}$ and $\overrightarrow{b}=\lambda \hat{i}+\hat{j}+2 \lambda \hat{k}$. If the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is greater than $90^{\circ}$, then $\lambda$ satisfies the inequality:

For $p > 0$,a vector $\vec{v}_{2} = 2 \hat{i} + (p + 1) \hat{j}$ is obtained by rotating the vector $\vec{v}_{1} = \sqrt{3} p \hat{i} + \hat{j}$ by an angle $\theta$ about the origin in the counter-clockwise direction. If $\tan \theta = \frac{(\alpha \sqrt{3} - 2)}{4 \sqrt{3} + 3}$,then the value of $\alpha$ is equal to $....$